[G] Mutalisk Bounce Damage & Muta vs Muta Upgrades

ARMOR
First of all, “ARMOR” in this thread refers to anything that reduces damages: Terran/Protoss armor/plating, Zerg carapace, Protoss shield, guardian shield etc.
Mimimum ARMOR is 0 of unupgraded marine, zergling, protoss shield etc.
Maximum ARMOR is 8 of max upgraded ultralisk/battlecruiser (6 ARMOR) inside guardian shield (+2 ARMOR).

Basic Damage Rule
+x attack upgraded mutalisk vs +y ARMOR units/structures
1st hit = (9 +x)/1 – y
2nd hit = (9+x)/3 – y
3rd hit = (9+x)/9 – y
Total = 13(9+x)/9 – 3y
Essentially, this 13/9 = 1.4444… < 3 is the reason why carapace is preferred in muta vs muta.

When glave wurm hits different units/structures with different ARMOR (+a, +b, +c)
1st hit = (9+x)/1 – a
2nd hit = (9+x)/3 – b
3rd hit = (9+x)/9 – c
As you can see, ARMOR value of the previous target doesn’t affect the following damages.

Upgrade Notation
2-1 mutalisk = +2 attack upgraded, +1 carapace upgraded mutalisk, not the other way around.
I know carapace is on the left side at the bottom of the screen, but at spire, attack upgrade is on the left. Casters usually call attack upgrade first for any unit, so it should make sense. When someone says “two one upgraded marines,” these marines almost always have +2 infantry attack and +1 infantry armor upgrades. Same rule applies.

Decimal Damage
exists in SC2. It seems that the number in the 5th decimal place is rounded DOWN to give a damage value to the 4th decimal places. For example, 2nd hit for 2-0 muta vs 2 ARMOR is (11/3 – 2) = 1.6666 damage. Not that it matters in many situations, but it is NOT 1.6667 nor 1.6666....(repeated infinitely) damage. In a test, doing this 1.6666 damage 60 times deals 99.9960 damage, which does NOT kill a 100 health ghost. (11/3 – 2) = 1.66660 in SC2 as far as I could tell. 1.6667 would have dealt 100.0020 damage and killed the ghost.

Minimum Damage
is 1
…
…
…
NO. Absolutely not. It is 0.5000 damage.
Whenever
{(Supposed Damage vs 0 ARMOR) – (# of ARMOR)}=< 0.5,
then the damage dealt = 0.5
0-0 marine vs 3-3 battlecruiser, SCV vs +2 structure armor upgraded PF etc. are easy ways to confirm. This fact plays a significant role in the 3rd hit of mutalisk glave wurm. Every 3rd hit except for vs 0 ARMOR becomes 0.5000 damage due to this. If the minimum damage were 1.0000 in SC2, then mutalisk would be a slightly better unit. (about 3~8% total DPS increase)

Purple is for +0 attack upgraded mutalisk vs 0 ARMOR
Tables must be self-explanatory.

Since double upgrade is standard in TvZ, there is not much to talk about. The most relevant fact here is that SCVs become significantly more resistant to muta harass when going bio rather than mech because of extra ARMOR for SCVs that only comes from infantry armor at Engineering Bay.
0-0 muta vs 0-1 SCV is 19.2% reduced damage from 0-0 SCV for mech builds.
1-0 muta vs 0-2 SCV is 24.4% reduced.
2-0 muta vs 0-3 SCV is 29.5% reduced.
You don’t need to choose bio solely for this reason, but when going mech, you have to be aware that SCVs are more vulnerable. This fact is often overlooked.

Protoss usually needs +3 attack upgrade anyways because +3 colossi can 1 shot zerglings no matter what carapace they have while +2 colossi can’t 1 shot +2/3 carapace zerglings. Armor upgrade vs shield upgrade argument is not as simple as some might think. I’d say that going armor is better than shield in general, but if I have to name a unit that shield upgrade fares relatively well against, it is mutalisk. This is primarily because mutalisk’s glave wurm is the only Zerg attack with potential minimum damage = 0.5000. Many Protoss units have natural armor, and extra armor can’t reduce the 3rd hit below the minimum while boosting shield from +0 to +1 helps vs all hits. Structures live longer in base race scenarios as well. Therefore, while shield upgrade is not necessarily better than armor upgrade overall, vs muta/ling is probably the best time to consider shield upgrade. Shield for air units such as phoenix etc. doesn't hurt either. More on Protoss armor vs shield in another thread when I have time.

Zerglings under muta vs muta battle etc. is irrelevant in this thread because both sides should have exactly the same outside conditions for comparison. It’s all about muta vs muta fighting each other.
Assuming,
1. Always Player A > Player B in terms of upgrade advantage.
2. Player A has 0 or 1 more upgrade than Player B, not 2 or higher. That is, 1-1 vs 0-0, 2-2 vs 0-1 etc. are not considered here because it shouldn’t happen in the first place and it’s not worth analyzing when Player B is massively behind in every aspect.
3. No hive tech spire upgrade. 2-2 is generally the goal in muta vs muta. 0-3, 1-3 etc. almost never happens.
then, there are 21 cases where Player A has an upgrade advantage over Player B:



Analysis:
1 upgrade vs 0 upgrade (Case 1 and 2)
A/B section is what’s important in the table above. You are trying to be Player A in most cases.
0-1 vs 0-0 = 124%
1-0 vs 0-0 = 111%
Carapace upgrade is more expensive by 50 mineral/gas, but having 0-1 is much better if the upgrade manages to finish.

1 upgrade vs 1 upgrade (Case 3)
0-1 vs 1-0 = 110%
is why no one should go for attack upgrade first, unless when trying to hit a weird timing with 50 extra gas. Even so, not upgrading at all to save 150 gas all together would be better for such a timing. Always carapace first has been the known fact for years, and this is why.

2 upgrades vs 1 upgrade (Case 4~9, especially 5 and 7)
Both players should choose 0-1 first for the reason explained above, so relevant ones are
0-2 vs 0-1 = 124%
1-1 vs 0-1 = 113%
Therefore, on paper, +2 carapace looks better than +1 attack for the second upgrade choice. However,
+2 carapace = 225 mineral/gas 190sec
+1 attack = 100 mineral/gas 160sec
means that +1 attack finishes 30sec earlier with at least 1 extra mutalisk. This is why going +1 attack after +1 carapace is the standard. More on why not +2 carapace next.

2 upgrades vs 2 upgrades (Case 10~12, especially 12)
0-2 vs 1-1 = 107%
125 extra gas for 1-1 player which translates into 1 more mutalisk makes this fight an interesting one. According my tests with unit editor with random a-moving mutalisk concave,
0-2 side never loses the following fights as it shouldn’t:
(100%) 20muta 0-2 vs 1-1 20muta (0%)
(100%) 25muta 0-2 vs 1-1 25muta (0%)
(100%) 30muta 0-2 vs 1-1 30muta (0%)
7% advantage seems to be good enough to overcome occasional inferior concave/clump.

However, because 1-1 side should have 1 more mutalisk, I ran the following tests as well (10 each), and the winrates are:
(20%) 20muta 0-2 vs 1-1 21muta (80%)
(30%) 25muta 0-2 vs 1-1 26muta (70%)
(60%) 30muta 0-2 vs 1-1 31muta (40%)
1 extra mutalisk = 3~5% more DPS +120 extra health negates 0-2 side’s 7% upgrade advantage very well. With these results plus the fact that +2 carapace takes 30sec longer to finish, I conclude that +1 attack upgrade should be the 2nd upgrade choice.

3 upgrades vs 2 upgrades (Case 13~18, especially 15 and 17)
1-2 vs 1-1 = 120%
2-1 vs 1-1 = 111%
As expected, carapace is better again just like 0-1 is preferred to 1-0. Saving 50gas at this point of the game is not a good move. Instead, save 225gas and hit timing or transition out of muta. 175gas for an inferior choice makes no sense. Get +2 carapace, or don’t upgrade any more.

3 upgrades vs 3 upgrades (Case 19)
1-2 vs 2-1 = 106%
This situation shouldn’t happen. Get carapace first, enough said.

4 upgrades vs 3 upgrades (Case 20 and 21)
2-2 vs 1-2 = 118%
2-2 vs 2-1 = 114%
Usually, the game is over before this point, or one side has already transitioned out of muta.

Conclusion:
The best upgrade order is
+1 Carapace
+1 Attack
+2 Carapace
+2 Attack

I doubt this is a surprise for anyone. We’ve known this since WoL. Attack first could be better in some rare situations where your muta is solely used to pick off drones, kill extractors and never engage vs opponent’s muta, but in like 95%+ of the time, carapace first is superior. Althought minimum damage mechanics makes it a bit more complicated, the core reason for this carapace advantage is because
1 + 1/3 + 1/9 = 13/9 =1.4444…. for attack
1 + 1 + 1 = 3 for carapace
and 1.4444…. < 3 as mentioned above.

Because pretty much everyone knows
+1 Carapace
+1 Attack
+2 Carapace
+2 Attack
order, muta battle with something like 2-0 vs 1-0 (Case 8) never occurs in real games except between players who have no clue what they are doing. When both players follow the standard upgrade path, this is how the game transitions:
0-0 vs 0-0 Even
0-1 vs 0-0 Case1
0-1 vs 0-1 Even
1-1 vs 0-1 Case 7
1-1 vs 1-1 Even
1-2 vs 1-1 Case 15
1-2 vs 1-2 Even
2-2 vs 1-2 Case 21
2-2 vs 2-2 Even
Player B with slower upgrade always catches up later with the same kind of upgrade to make it even again. The length of each period depends on situations in game. Sometimes, Player B surpasses in upgrades, but the transition itself is virtually the same.



The graph shows that when you are ahead in upgrades, it is better to engage in a muta v muta battle after finishing a carapace upgrade instead of an attack upgrade. The reason is twofold. One is simply having even better advantage (120%+) rather than 110%+. Another reason is that after finishing a carapace upgrade, the next upgrade choice is an attack upgrade which is a cheaper investment that doesn’t hurt the temporary mutalisk count as much. 1-1 vs 0-1 = 113% timing attempt is still nice, but 225gas investment to continue upgrading is a big burden compared to 0-1 vs 0-0 =124% with 100gas for the next upgrade. There are so many other factors in game, so it’s not be-all and end-all, yet it is especially worth noting that
1-1 vs 0-1 = 113% advantage is not as big as
0-1 vs 0-0 = 124% and
1-2 vs 1-1 = 120%
If possible, forcing engagement @ 0-1 or waiting until 1-2 is better than pushing smaller advantage @1-1 as a player with faster upgrade.

Original research was done even before I signed up on TL a year ago. So, these damage numbers are from WoL. Nevertheless, damage mechanics are the same, and faster mutalisk regeneration only means that battles last longer. I didn’t make a thread back then because mutalisk play was slowly dying a year ago in favor of infestor in ZvT and roach in ZvZ/P. With HotS, mutalisk is coming back and ZvZ is all about muta vs muta for now, so I thought this would be a good time to revisit my old work. Because of my internet issue, I haven’t been able to play HotS yet. That’s why I decided to spend some more time on my articles. I already have several new research ideas for HotS, but unfortunately, I have to delay them for a while. In the meantime, an article without HotS requirement is coming up. I have to be glad that I can entertain myself without actually playing the game Thank you for reading. I hope this thread was helpful.